sophie (sonia) schirmer

Landscapes & Algorithms for Quantum Control

Sophie (Sonia) Schirmer

Dept of Applied Maths & Theoretical PhysicsUniversity of Cambridge, UK

April 15, 2011

Sophie (Sonia) Schirmer Landscapes & Algorithms for Quantum Control

Control Landscapes: What do we really know?

“Kinematic control landscapes” generally universally nice

1 Pure-state transfer problems (finite-dim.)

Only global extrema, no saddles

2 Density-matrix/observable optimization problems

Saddles but no suboptimal extrema

3 Unitary operator optimization: depends on domain

U(N): Critical manifolds but no trapsSU(N): Beware of root-of-unity trapsPU(N): Critical manifolds but no traps

Actual control landscapes — things get messy


Kinematic vs Actual Control Landscape

Idea: Decompose map from control function spaceF : L2[0,T ] 7→ IR into parts

1 Uf : L2[0,T ] 7→ U(N) and2 G : U(N) 7→ IR (easy problem)

If we are lucky, maybe we can study the control landscape of(2) — kinematic control landscape — and apply the results tothe actual (hard) problem (1).

More mathematically precisely: If (1) is(a) surjective (equivalent to controllability) and(b) has maximum rank everywhere

then the actual control landscape looks like the kinematic one.

But are we this lucky?


Assumption (a) – controllability

Definition: Given the control system H = H0 + f (t)H1 let L0 bethe set of commutator expression in iH0 and iH1 joined with iH1;L = L0 ∪ iH0. G0 = exp(L0) and G = exp(L).

Theorem (Controllability)

If L = su(N) or u(N) then there exists a time Tmax andneighborhood N of (−iH0,−iH1) in L×L such that for all s ∈ Nand g ∈ G there is a control taking s to g in some timeT < Tmax.

Theorem (Exact-time controllability)

If L0 = su(N) or u(N) then there is a critical time Tc andneighborhood N of (−iH0,−iH1) ∈ L×L such that for all s ∈ N ,g ∈ G and T > Tc there is a control taking s to g in time T .

Very many quantum systems satisfy L0 = su(N) or u(N).


Assumption (b) – no singular points

1 Gate optimization: assumption (b) never satisfied overany function space containing constant controls as all ofthese rank-deficient; many non-constant controls also failthis test.

2 State transfer/observable optimization: for every bilinearHamiltonian control system there exist pairs of initial andtarget states for which there are singular controls.

Failure of (b) implies that actual landscape need notresemble kinematic one — Further analysis needed!

1 Counter-examples show that suboptimal extrema exist forall types of problems above [arXiv:1004.3492]

2 Many critical points with semi-definite Hessian 2nd-orderpotential traps [PRL 106,120402]

Are potential traps always actual traps?


Control Landscapes: What is a Trap?

Not a trivial question!

1 Is it a local extremum for which fidelity (error) assumesvalue less than global maximum (minimum)?

2 Is it any point that can attract trajectories?

(2) includes saddles that have domains of attraction but in anyneighborhood there are also points not attracted to saddle.

3 Cases: domain of attraction1 contains a neighborhood of the point – the case for strict

local extrema under the usual assumptions – typical traps2 within any neighborhood of the point is open (has positive

measure) but not everything3 for the point is lower dimensional or has empty interior

(measure-zero)


Simple example: f (x , y) = x2 + αyn, n ≥ 2

(0,0) critical point: fx(0,0) = 2x = 0, fy = nαyn−1 = 0.

Hessian:

H(x , y) =

(2 00 αn(n − 1)yn−2

)

H(0,0) > 0 (positive definite) if n = 2 and α > 0

H(0,0) ≥ 0 (positive semi-definite) for n > 2.

Definition: ”Second-order” trap: Hessian H(0,0) ≥ 0

Which of the following are traps?


Minimum, Hessian positive definite

−6−4

−20

24

6

−5

0

5

0

20

40

60

80

xy

minimum at (0,0) ; H(0,0) > 0


Saddle, Hessian indefinite

−5

0

5

−5

0

5

−40

−20

0

20

40

x2−y2

xy

saddle at (0,0) ; H(0,0) indef


Second-order trap – minimum

−5

0

5

−6−4

−20

24

60

500

1000

1500

2000

x

x2+y4

y

minimum at (0,0) ; H(0,0) ≥ 0


2nd-order trap – saddle / pos. measure attractive set

−1−0.500.51−1

−0.50

0.51

−1

−0.5

0

0.5

1

1.5

2

y

x2−y3

x

2nd order trap: Saddle with pos. measure domain of attraction


So what is a trap now?

1 Not all “second-order” traps are extrema – many saddles.

Semi-definite Hessian not sufficient for extremum.

Second-order traps that are saddles exist for quantumcontrol problem — see Example 2 in arXiv:1004.3492.

Could argue saddles are not strictly traps as in anyneighborhood some points will not converge to the saddle.

2 But saddles have domains of attraction; second-ordertraps can have positive measure domains of attraction.

3 Domains of attraction depend on algorithm!

Local extrema traps for all local optimization algorithms –how many second-order traps are suboptimal extrema?

Convergence to saddles possible, especially if they havepositive measure domains of attraction!


Domains of attraction of a saddle

-2.0

-1.5

-1.0

-0.5

0.00.5

1.01.5

-3

-2

-1 0 1 2 3

Trapping regions fordifferent optimizationalgorithms:

green: steepest descent(concurrent)

red: Newton (concurrent)

blue: sequential

What is the probability of trapping?How can we detect when an optimization run is going badly?


Convergence: The Good, The Bad & The Ugly

0 100 200 300 400 50010

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

iteration number (n)

1−F

id(n

) of

100

suc

cess

ful r

uns

0 1000 2000 3000 4000 5000 6000 7000 8000 900010

−3

10−2

10−1

100

101

iteration number n

1−F

id(n

)

0 50 100 150 20010

−4

10−3

10−2

10−1

100

iteration n

1−F

id(n

)

intermittant trapping

0 500 1000 1500 2000 250010

−4

10−3

10−2

10−1

100

iteration n

Err

orbad runs: convergence stalls at 98%

successful runs reach target fidelity


Optimization Algorithms: Which perform best?

Numerical optimization expensive hence efficiency is key.

Many algorithms, many choices — How do we choose?

1 Global vs Local Optimization:Optimization algorithms designed to find local extrema farmore efficient than global optimization strategiesLocal optimization algorithms preferable if probability oftrapping low.

2 Sequential vs concurrent update:Sequential update (Krotov) inspired by dynamical systems:discrete version of continuous flow (Krotov)Concurrent update motivated by conventional optimization

3 How to choose effective update rules?Gradient vs higher-order methods (Newton/quasi-Newton)Search length adjustment (line search)

4 Discretization: How to parametrize the controls?


Preliminary observations

1 Local optimization algorithms outperform global ones ingeneral — trapping probability low for suitable initial fields

2 Far from a global optimum sequential update algorithmsgenerally help you escape faster — but only up to a point.

Sequential update allow larger local changes of the fieldsbut rapid initial gains may be paid for in terms of decreasedasymptotic rate of convergence.

3 Close to the top concurrent update has clear edge as it canexploit nonlocal temporal correlations of the fields.

4 Discretization must be considered in analysis — affectsGradient accuracy — don’t rely on continuous limitEffective line search strategies — use quadratic modelRate of convergence — don’t be too greedyChoice of penalty terms — regularizing penalty termsunnecessary for finite time steps


Algorithm Comparison

0 100 200 300 400 500 600 700 80010

−4

10−3

10−2

10−1

100

X: 100.6Y: 0.003186

computational time (sec)

med

ian

erro

r (

succ

essf

ul r

uns)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

computational time (sec)m

edia

n er

ror

Newton (seq)

gradient v3conc (BFGS)

gradient (greedy)

gradient v1

Concurrent (BFGS)

Best sequential update


Summary and Conclusions

1 Control Landscape:Actual dynamic landscape appears to be far richer thankinematic analysis suggests — many questions

2 Numerical Algorithms for solving opt. control problems

Progress in understanding convergence behaviour, etc. butsignificant room for improvement!

Many questions — e.g., are some algorithms more likely toget trapped in (higher-order) saddles?

3 Parametrization of controls/discretization crucialAnalysis of infinite dimensional control problem may notactually tell us much about what happens in discrete case.

4 Robustness: Optimal control solutions naturally robust butsome perturbations that are more detrimental than others.


sophie (sonia) schirmer

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